Sphere of influence (astrodynamics)

A sphere of influence (SOI) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects (such as moons), despite the presence of the much more massive (but distant) Sun. In a more general sense, the patched conic approximation is only valid within the SOI.

The general equation describing the radius of the sphere $r_{SOI}$ of a planet:

${\displaystyle r_{SOI}\approx a\left({\frac {m}{M}}\right)^{2/5}}$

where

$a$ is the semimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).

$m$ and $M$ are the masses of the smaller and the larger object (usually a planet and the Sun), respectively.

In the patched conic approximation, once an object leaves the planet's SOI, the primary/only gravitational influence is the Sun (until the object enters another body's SOI). Because the definition of r_SOI relies on the presence of the Sun and a planet, the term is only applicable in a three-body or greater system and requires the mass of the primary body to be much greater than the mass of the secondary body. This changes the three-body problem into a restricted two-body problem.

Table of selected SOI radii relative to the Sun

Body	SOI radius (10^6 km)	SOI radius (body radii)
Mercury	0.112	46
Venus	0.616	102
Earth	0.924	145
Moon	0.0661	38
Mars	0.576	170
Jupiter	48.2	687
Saturn	54.6	1025
Uranus	51.8	2040
Neptune	86.8	3525

Increased accuracy on the SOI

The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance $\theta$ from the massive body. A more accurate formula is given by

${\displaystyle r_{SOI}(\theta )\approx a\left({\frac {m}{M}}\right)^{2/5}{\frac {1}{\sqrt[{10}]{1+3\cos(\theta )^{2}}}}}$

Averaging over all possible directions we get

${\displaystyle {\overline {r_{SOI}}}=0.9431a\left({\frac {m}{M}}\right)^{2/5}}$

See also

• Hill sphere
• Sphere of influence (black hole)

References

• Bate, Roger R.; Donald D. Mueller; Jerry E. White (1971). Fundamentals of Astrodynamics. New York: Dover Publications. pp. 333–334. ISBN 0-486-60061-0.

• Sellers, Jerry J.; Astore, William J.; Giffen, Robert B.; Larson, Wiley J. (2004). Kirkpatrick, Douglas H., ed. Understanding Space: An Introduction to Astronautics (2nd ed.). McGraw Hill. pp. 228, 738. ISBN 0-07-294364-5.

• Danby, J. M. A. (2003). Fundamentals of celestial mechanics (2. ed., rev. and enlarged, 5. print. ed.). Richmond, Va., U.S.A.: Willmann-Bell. pp. 352–353. ISBN 0-943396-20-4.
• Project Pluto